## Surface waves and volume waves in a photonic crystal slab

Optics Express, Vol. 15, Issue 13, pp. 7913-7921 (2007)

http://dx.doi.org/10.1364/OE.15.007913

Acrobat PDF (2746 KB)

### Abstract

The volume and surface waves supported by a 2D PC slab with termination condition are systematically studied using the rigorous mode-matching method incorporating the Floquet’s solutions. It is interesting to observe that the surface waves are caused by the perturbation of the PC-slab modes from the imposed termination condition, enabling the transition from volume wave to surface wave. The perturbed dispersion curves and electric field strength distribution over the structure are drawn together with the unperturbed ones (without termination condition) to identify the type of bound waves.

© 2007 Optical Society of America

## 1. Introduction

1. V. G. Veselago, “The electrodynamics of substance with simultaneously negative values of ε and μ,” Sov. Phy. Usp. **10**, 509–514 (1968). [CrossRef]

**k**,

**E**and

**H**form a left-handed set of vectors. A striking observation of superlensing using a uniform medium with negative permittivity and permeability was found by Pendry [2

2. J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. **85**, 3966 (2000). [CrossRef] [PubMed]

3. X. S. Rao and C. K. Ong, “Amplification of evanescent waves in a lossy left-handed material slab,” Phys. Rev. B **68**, 113103 (2003). [CrossRef]

4. X. S. Rao and C. K. Ong, “Subwavelength imaging by a left-handed material superlens,” Phys. Rev. B **68**, 067601 (2003). [CrossRef]

5. Chiyan Luo, Steven G. Johnson, J. D. Joannopoulos, and J. B. Pendry, “Subwavelength imaging in photonic crystal,” Phys. Rev. B **68**, 045115 (2003). [CrossRef]

6. Sanshui Xiao, Min Qiu, Zhichao Ruan, and Sailing He, “Influence of the surface termination to the point imaging by a photonic crystal slab with negative refraction,” Appl. Phys. Lett. **85**, 4269 (2004). [CrossRef]

7. Esteban Moreno, F. J. Garcia-Vidal, and L. Martin-Moreno, “Enhanced transmission and beaming of light via photonic crystal surface modes,” Phys. Rev. B **69**, 121402 (2004). [CrossRef]

8. P. Kramper, M. Agio, C. M. Soukoulis, A. Birner, F. Müller, R. B. Wehrspohn, U. Gösele, and V. Sandoghdar, “Highly Directional Emission from Photonic Crystal Waveguides of Subwavelength Width,” Phys. Rev. Lett. **92**, 113909 (2004). [CrossRef]

9. X. Wang and K. Kempa, “Effects of disorder on subwavelength lensing in two-dimensional photonic crystal slabs,” Phys. Rev. B **71**, 085101 (2005). [CrossRef]

10. J. Bravo-Abad, F. J. García-Vidal, and L. Martín-Moreno, “Resonant Transmission of Light Through Finite Chains of Subwavelength Holes in a Metallic Film,” Phys. Rev. Lett. **93**, 227401 (2004). [CrossRef] [PubMed]

11. R. Moussa, Th. Koschny, and C. M. Soukoulis, “Excitation of surface waves in a photonic crystal with negative refraction: The role of surface termination,” Phys. Rev. B **74**, 115111 (2006). [CrossRef]

13. T. Tamir and S. Zhang, “Modal Transmission-Line Theory of Multilayered Grating Structures,” IEEE J. Lightwave Technol. **14**, 914 (1996). [CrossRef]

14. Ruey Bing Hwang and Cherng Chyi Hsiao, “Frequency-selective transmission by a leaky parallel-plate-like waveguide,” IEEE Trans. on Antennas and Propagation **54**, 121 (2006). [CrossRef]

## 2. Method of Analysis

*x*- and

*y*- direction, while it is finite the

*z*direction. Below the periodic structure at a distance

*h*, there is an OC (open-circuit) or a SC (short-circuit) termination. Since the electromagnetic fields as well as the structure are assumed to be invariant along the

*y*direction, the problem can be individually treated as

*E*- and

_{y}*H*- mode. In this paper, we only considered the condition of

_{y}*E*polarization.

_{y}*z*direction). The scattering characteristic of plane wave by a 1D periodic structure can be considered as a basic building block for constructing the scattering characteristic of the overall structure. At first, the plane-wave expansion method is employed to solve the eigen-solutions in an infinite 1D periodic medium. After matching the boundary conditions at the interface between the 1D periodic medium and uniform surrounding medium, the input-output relation of the 1D periodic layer was obtained. By cascading the input-output relation of each 1D periodic layer, the scattering characteristic including the reflection- and transmission- efficiency of each space harmonic can be determined accordingly. Moreover, the dispersion relation of the modes (eigen-solutions of a source-free problem) can be determined by solving the transverse resonance equation. Since the detail mathematical procedures for resolving the dispersion relation of bound waves and the band structure could be found in literature [12–15

12. S. Enoch, G. Tayeb, and B. Gralak, “The richness of the dispersion relation of electromagnetic bandgap materials,” IEEE Trans. on Antennas and Propagation **51**, 2659 (2003). [CrossRef]

## 3. Numerical Results

*x*and

*z*directions (

*d*and

_{x}*d*) are 0.5

_{z}*d*, 1.0

*d*and 1.0

*d*, respectively, where

*d*is the period along the

*x*and

*z*directions. The number of 1D periodic layer is 5, whereas the number of periods along the

*x*direction is infinity. The distance from the 2D PC to the termination plane denotes

*h*.

*d*/

_{x}*λ*) and normalized propagation constant along the

*x*direction (

*β*/2

_{x}d_{x}*π*), respectively. The straight line with unity slope is the light line (

*β*=

_{x}*k*); the region with slop greater than unity represents the slow-wave (or bound-wave) region where the electromagnetic waves are confined within the PC slab, whereas the region with slope smaller than unity is the fast-wave region where the electromagnetic waves may radiate into the surrounding medium. The wave guiding characteristics in defect region and its leaky-wave phenomena were well known in literature [13

_{o}13. T. Tamir and S. Zhang, “Modal Transmission-Line Theory of Multilayered Grating Structures,” IEEE J. Lightwave Technol. **14**, 914 (1996). [CrossRef]

14. Ruey Bing Hwang and Cherng Chyi Hsiao, “Frequency-selective transmission by a leaky parallel-plate-like waveguide,” IEEE Trans. on Antennas and Propagation **54**, 121 (2006). [CrossRef]

*projected band-structure*of the corresponding infinite 2D PC was drawn [15]; the region drawn in yellow color is the pass-band region, while the region in white color is the stop-band region. The two different termination conditions are taken into account in this numerical example, which are open-circuit and without termination (infinite in extent of the uniform medium), respectively. The curves in black and red colors represent the dispersion characteristics of waveguide modes for the case of open-circuit termination and without termination, respectively. These dispersion curves are very similar to those of the bound waves in multiple uniform dielectric layers, excluding the band-gap on each dispersion curve resulted from the contra-flow coupling between the

*x*-direction space harmonics. Since the structure contains five 1D periodic layers, there exist five dispersion curves of the fundamental mode shown in this figure.

*h*=0.01) was imposed. Obviously, such a perturbation on the PC slab enables the dispersion curves of un-terminated case to move downwards and go away the pass-band region. The dispersion curve in the stop-band region represents that the electromagnetic field experiences a strong reflection in the 2D PC. Moreover, since the

*x*-direction phase constant (

*β*) is greater than the free-space wave number (

_{x}*k*), the wave is decaying in the surrounding medium. Therefore, this wave is decaying along both directions from the interface, at

_{o}*z*= 0, between 2D PC and uniform medium. To demonstrate it, we plot the distribution of the electric field strength (magnitude of

*E*component) over the structure at the normalized frequency

_{y}*d*/λ = 0.25. From Fig. 3, we observe that the field is decaying along the 2D PC region; however, the decaying below the

_{x}*z*= 0 region is not easy to observe because that the termination plane is very close to the interface (

*h*=0.01).

*h*=0.01, shown in Fig. 4. From this figure, it is obviously to see that, contrary to the OC termination, the dispersion curves of the SC termination (in black color) move upwards; for example, the one in the stop-band region moves out of the pass-band region. Although not shown here, the distribution of the electric field intensity of the dispersion curves in the stop-band region also exhibits exponential decay in the 2D PC structure.

*h*to 0.1 to calculate the dispersion curves. As shown in fig. 5, the dispersion curves of the OC termination still moves downwards; however, the displacement away from the un-terminated case decreases in comparison with the results shown in Fig. 2. It indicates that the influence due to the OC termination is decreasing as the termination distance

*h*is increasing. To verify this phenomenon, we have consecutively increased the distance

*h*from 0.1 to 0.245 to see the variation on the displacement between the cases of OC termination and without termination. Although not shown here, the results indicate that the increasing in the termination distance, the two sets of dispersion curves will gradually approach to each other. Moreover, we found that as the termination distance

*h*is greater than 0.245, the two group of dispersion curves are hard to be distinguished. All the dispersion curves are volume waves and stay in the pass band region.

*h*=0.245, it is apparently to see that the two groups of dispersion curves are close to each other. The physical interpretation is given below. Since the bound waves have relative refractive index (

*β*/

_{x}*k*) greater than unity, they are exponentially decaying in the uniform surrounding medium. If the termination plane is far away from the 2D PC, the wave is decaying considerably at the position of the termination plane; and the reflected wave is, of course, insignificant. Therefore, such a situation is very similar to the case without termination, and the perturbation on the dispersion curve is negligible. Conversely, when the termination plane is close to the 2D PC, the waves have not yet decayed at the termination plane, thus the reflected waves is appreciable. The dispersion curves should be altered by the incorporation of the reflected waves.

_{o}*d*/λ = 0.25. As was shown in Fig. 6, at this normalized frequency, there exist three bound waves for each termination condition, respectively, where the normalized propagation constant along the

_{x}*x*direction (

*β*/ko) are 1.1972, 1.3872 and 1.4779 for the OC termination, and are 1.1542, 1.3449 and 1.4532 for the case without termination, respectively. Figures 7, 8 and 9 demonstrate the field strength distribution (magnitude of

_{x}*E*component) of the three modes for the two termination conditions, respectively. Excluding the third mode in the OC termination case, the the

_{y}*x*-direction normalized propagation constant are all in the pass-band region and belong to volume waves. Namely, the field can penetrate into the 2D PC structure while it decays in the uniform surrounding medium. Since the first two normalized propagation constants of respective termination conditions locate in the pass-band region and are close to each other, they are apparently to have the similar field distributions as expected from physical intuition.

*z*= 0, confirming its wave-propagation behavior in stop-band region. Unlike the picture shown in the left-hand side, the bound wave of the un-terminated case (the third mode) has maximum electric-field strength around the center of the structure. Notably, since this mode is centered at the termination plane, it does not allow to visualize the surface state.

## 4. Conclusion

## Acknowledgments

## References and links

1. | V. G. Veselago, “The electrodynamics of substance with simultaneously negative values of ε and μ,” Sov. Phy. Usp. |

2. | J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. |

3. | X. S. Rao and C. K. Ong, “Amplification of evanescent waves in a lossy left-handed material slab,” Phys. Rev. B |

4. | X. S. Rao and C. K. Ong, “Subwavelength imaging by a left-handed material superlens,” Phys. Rev. B |

5. | Chiyan Luo, Steven G. Johnson, J. D. Joannopoulos, and J. B. Pendry, “Subwavelength imaging in photonic crystal,” Phys. Rev. B |

6. | Sanshui Xiao, Min Qiu, Zhichao Ruan, and Sailing He, “Influence of the surface termination to the point imaging by a photonic crystal slab with negative refraction,” Appl. Phys. Lett. |

7. | Esteban Moreno, F. J. Garcia-Vidal, and L. Martin-Moreno, “Enhanced transmission and beaming of light via photonic crystal surface modes,” Phys. Rev. B |

8. | P. Kramper, M. Agio, C. M. Soukoulis, A. Birner, F. Müller, R. B. Wehrspohn, U. Gösele, and V. Sandoghdar, “Highly Directional Emission from Photonic Crystal Waveguides of Subwavelength Width,” Phys. Rev. Lett. |

9. | X. Wang and K. Kempa, “Effects of disorder on subwavelength lensing in two-dimensional photonic crystal slabs,” Phys. Rev. B |

10. | J. Bravo-Abad, F. J. García-Vidal, and L. Martín-Moreno, “Resonant Transmission of Light Through Finite Chains of Subwavelength Holes in a Metallic Film,” Phys. Rev. Lett. |

11. | R. Moussa, Th. Koschny, and C. M. Soukoulis, “Excitation of surface waves in a photonic crystal with negative refraction: The role of surface termination,” Phys. Rev. B |

12. | S. Enoch, G. Tayeb, and B. Gralak, “The richness of the dispersion relation of electromagnetic bandgap materials,” IEEE Trans. on Antennas and Propagation |

13. | T. Tamir and S. Zhang, “Modal Transmission-Line Theory of Multilayered Grating Structures,” IEEE J. Lightwave Technol. |

14. | Ruey Bing Hwang and Cherng Chyi Hsiao, “Frequency-selective transmission by a leaky parallel-plate-like waveguide,” IEEE Trans. on Antennas and Propagation |

15. | J. D, Joannopolous, R. D. Meade, and J. N. Winn, |

**OCIS Codes**

(230.7370) Optical devices : Waveguides

(260.2110) Physical optics : Electromagnetic optics

**ToC Category:**

Photonic Crystals

**History**

Original Manuscript: March 26, 2007

Revised Manuscript: May 22, 2007

Manuscript Accepted: June 6, 2007

Published: June 11, 2007

**Citation**

Raybeam Hwang, "Surface waves and volume waves in a photonic crystal slab," Opt. Express **15**, 7913-7921 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-13-7913

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### References

- V. G. Veselago, "The electrodynamics of substance with simultaneously negative values of ε and μ," Sov. Phy. Usp. 10, 509-514 (1968). [CrossRef]
- J. B. Pendry, "Negative refraction makes a perfect lens," Phys. Rev. Lett. 85, 3966 (2000). [CrossRef] [PubMed]
- X. S. Rao and C. K. Ong, "Amplification of evanescent waves in a lossy left-handed material slab," Phys. Rev. B 68, 113103 (2003). [CrossRef]
- X. S. Rao and C. K. Ong, "Subwavelength imaging by a left-handed material superlens," Phys. Rev. B 68, 067601 (2003). [CrossRef]
- Chiyan Luo, Steven G. Johnson, J. D. Joannopoulos and J. B. Pendry, "Subwavelength imaging in photonic crystal," Phys. Rev. B 68, 045115 (2003). [CrossRef]
- Sanshui Xiao, Min Qiu, Zhichao Ruan, and Sailing He, "Influence of the surface termination to the point imaging by a photonic crystal slab with negative refraction," Appl. Phys. Lett. 85, 4269 (2004). [CrossRef]
- Esteban Moreno, F. J. Garcia-Vidal, and L. Martin-Moreno, "Enhanced transmission and beaming of light via photonic crystal surface modes," Phys. Rev. B 69, 121402 (2004). [CrossRef]
- P. Kramper, M. Agio, C. M. Soukoulis, A. Birner, F. Müller, R. B. Wehrspohn, U. Gösele, and V. Sandoghdar, "Highly directional emission from photonic crystal waveguides of subwavelength width," Phys. Rev. Lett. 92, 113909 (2004). [CrossRef]
- X. Wang and K. Kempa, "Effects of disorder on subwavelength lensing in two-dimensional photonic crystal slabs," Phys. Rev. B 71, 085101 (2005). [CrossRef]
- J. Bravo-Abad, F. J. García-Vidal, and L. Martín-Moreno, "Resonant transmission of light through finite chains of subwavelength holes in a metallic film," Phys. Rev. Lett. 93, 227401 (2004). [CrossRef] [PubMed]
- R. Moussa, Th. Koschny and C. M. Soukoulis, "Excitation of surface waves in a photonic crystal with negative refraction: The role of surface termination," Phys. Rev. B 74, 115111 (2006). [CrossRef]
- S. Enoch, G. Tayeb and B. Gralak, "The richness of the dispersion relation of electromagnetic bandgap materials," IEEE Trans. on Antennas and Propagation 51, 2659 (2003). [CrossRef]
- T. Tamir and S. Zhang, "Modal transmission-line theory of multilayered grating structures," IEEE J. Lightwave Technol. 14, 914 (1996). [CrossRef]
- Ruey Bing Hwang and Cherng Chyi Hsiao, "Frequency-selective transmission by a leaky parallel-plate-like waveguide," IEEE Trans. on Antennas and Propagation 54, 121 (2006). [CrossRef]
- J. D, Joannopolous, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light, (Princeton, NJ: Princeton University Press, 1995).

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